{\LARGE \bfseries COM-303 - Signal Processing for Communications \\ Final Exam} \\
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{\large Saturday, June 21 2014, 09:15 to 12:15}
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\begin{itemize}
\item {\bf Write your name} on the top left corner of {\bf ALL sheets you turn in}, including this one. When you are done, \textbf{staple} all your sheets together \textbf{with this sheet on top}!
\item You can have two A4 sheet of \emph{handwritten} notes (front and back). Please \textbf{no photocopies, no books and no electronic devices}. Turn off your phone if you have it with you.
\item There are 5 problems for a total of 100 points; the number of points for each problem is indicated next to it.
\item Please write your derivations clearly, as there is partial credit.
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\begin{exercise}{(10 points)}
Consider the discrete-time signal $x[n] = \sinc(an)$ with $0 < a < 1$; compute the following sums:
{\em the impulse response of an ideal lowpass filter with cutoff frequency $\omega_c$ is }
\[
h[n] = (\omega_c/\pi)\sinc((\omega_c/\pi)n)
\]
{\em therefore $x[n]$ is the impulse response of an ideal lowpass filter with cutoff frequency $a\pi$, scaled by $1/a$ so that} $X(e^{j\omega}) = (1/a)\rect(\omega/(2a\pi))$. {\em From this:}
{\em A first order section with a pole in $z = \alpha$ has a transfer function $G(z) = 1/(1 - \alpha z^{-1})$ and impulse response $g[n] = \alpha^n u[n]$. Therefore the impulse response of the above system is}
{\em The transfer function of the system is} $H(z) = H_1(z)H_2(z)$ {\em where}
\[
H_{1,2}(z) = \frac{1}{1 - (1\pm j)z^{-1}}.
\]
{\em The system has therefore no zeros and two poles at $z = (1\pm j)$ or, in polar coordinates, at $z = e^{\pm j\frac{\pi}{4}}$ (note that the filter is not stable); its frequency response in magnitude follows the classic resonator pattern:}
Bellanger's Approximation is an empirical formula used to estimate the order of an equiripple lowpass filter based on its design specifications. For a lowpass filter with transition band $[\omega_p,\ \omega_s]$ and error tolerances of $\delta_p$ and $\delta_s$ in passband and stopband respectively, the filter order is going to be approximately
\[
N \approx \frac{-2\log_{10}(10\delta_p\delta_s)}{3(\omega_s - \omega_p)/2\pi} - 1
\]
Since the order is inversely proportional to the width of the transition band, ``sharp'' filters (i.e., filters with a narrow transition band) will require a lot of multiplications per output sample. The following questions will ask you to analyze an alternative design strategy called IFIR (Interpolated FIR), used to obtain sharp filters at a lower computational cost.
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To begin with, assume you have designed an $N$-tap FIR lowpass $H(z)$ with the following magnitude response (we're showing just the positive frequencies and neglecting the ripples):
\item describe filter $I(z)$ so that the cascade $G(z)I(z)$ is equivalent to a simple lowpass filter \\ \\
{\em $I(z)$ should be a lowpass filter that removes the high frequency image introduced by the upsampling. }
\item specify the passband and stopband frequencies of the global filter implemented by the cascade \\ \\
{\em the global filter is a lowpass with band edges:}
\begin{align*}
\omega_p &= \phi_p/2 \\
\omega_s &= \phi_s/2
\end{align*}
\item find the passband and stopband frequencies of $I(z)$ so as to maximize its transition band \\ \\
{\em to minimize the computational cost of the cascade we can keep the transition band as wide as possible. We could use the following values, for instance:
\begin{align*}
\theta_p &= \phi_p/2 \\
\theta_s &= \pi - \phi_s/2 \\
\end{align*}
for a transition band of $\Delta_I = \pi - (\phi_s + \phi_p)/2$.}
Consider now the following specifications for a lowpass filter:
\begin{align*}
\omega_p &= 0.3\pi \\
\omega_s &= 0.31\pi \\
\delta_p &= \delta_s = 0.01
\end{align*}
and compare a direct FIR with an IFIR implementation.
\begin{enumerate}
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\item estimate the order of a direct equiripple implementation of the filter using Bellanger's formula \\
{\em
\begin{align*}
N &\approx \frac{-2\log_{10}(10\cdot 10^{-2}\cdot 10^{-2})}{3(0.31-0.3)\pi/2\pi} - 1 \\
&= \frac{6}{0.015} - 1 = 399
\end{align*}
}
\item now consider an IFIR implementation: give the specifications for the initial IFIR filter $H(z)$ (i.e. the values of $\phi_p$ and $\phi_s$ to use in the design of $H(z)$) \\ \\
{\em the initial filter has double the passband and stopband frequencies, i.e.
\begin{align*}
\phi_p &= 0.6\pi \\
\phi_s &= 0.62\pi \\
\end{align*}
}
\item estimate the order of an equiripple implementation of $H(z)$ \\
\[
N_H \approx \frac{6}{0.03} - 1 = 199
\]
\item assume an optimal equiripple design for $I(z)$ using the maximum transition band $\Delta_I$ you found before and using $\delta_p = \delta_s = 0.01$; estimate the order of $I(z)$ \\ \\